Problem
Research questions and friction points this paper is trying to address.
Temporal Networks
Information Flow
Transitivity
The Complexity of Transitively Orienting Temporal Graphs
🤖 AI Summary
This paper investigates transitivity modeling of information flow in temporal networks, focusing on orienting time-labeled undirected edges to enforce transitivity constraints over time-non-decreasing (temporal transitive orientation) or strictly time-increasing (strictly temporal transitive orientation) directed paths. We formally define both notions and reveal their fundamental computational distinction: the former is polynomial-time decidable, whereas the latter is NP-complete. Leveraging combinatorial graph theory and temporal graph modeling, we design the first polynomial-time algorithm for deciding temporal transitive orientability. Via a carefully constructed reduction, we prove the NP-completeness of the strict variant. Furthermore, we systematically characterize the algorithmic tractability and computational hardness of related problems—including temporal transitive completion. Our results establish a theoretical foundation and provide essential algorithmic tools for modeling controllable information flow in dynamic networks.
📝 Abstract
In a temporal network with discrete time-labels on its edges, entities and information can only ``flow'' along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge $e={u,v}$ with time-label $t$ specifies that ``$u$ communicates with $v$ at time $t$''. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior. An orientation of a temporal graph is called temporally transitive if, whenever $u$ has a directed edge towards $v$ with time-label $t_1$ and $v$ has a directed edge towards $w$ with time-label $t_2geq t_1$, then $u$ also has a directed edge towards $w$ with some time-label $t_3geq t_2$. If we just demand that this implication holds whenever $t_2>t_1$, we call the orientation strictly temporally transitive, as it is based on the strict directed temporal path from $u$ to $w$. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph $mathcal{G}$ is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether $mathcal{G}$ is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Temporal Causality
Temporal Transitivity
Directional Information Flow
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